1. **State the problem:** We need to divide the polynomial $2x^{3}+3x^{2}+5x-8$ by $x^{2}+2x+3$. 2. **Recall the formula:** Polynomial division is similar to long division with numbers. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat. 3. **Divide the leading terms:** $\frac{2x^{3}}{x^{2}}=2x$. 4. **Multiply the divisor by $2x$:** $2x(x^{2}+2x+3)=2x^{3}+4x^{2}+6x$. 5. **Subtract this from the dividend:** $$\left(2x^{3}+3x^{2}+5x-8\right)-\left(2x^{3}+4x^{2}+6x\right) = \cancel{2x^{3}}+3x^{2}+5x-8 - \cancel{2x^{3}} -4x^{2} -6x = (3x^{2}-4x^{2}) + (5x-6x) -8 = -x^{2} - x -8$$ 6. **Repeat the division with the new polynomial:** Divide the leading term $-x^{2}$ by $x^{2}$ to get $-1$. 7. **Multiply the divisor by $-1$:** $-1(x^{2}+2x+3) = -x^{2} - 2x - 3$. 8. **Subtract this from the current polynomial:** $$(-x^{2} - x -8) - (-x^{2} - 2x - 3) = \cancel{-x^{2}} - x - 8 - \cancel{-x^{2}} + 2x + 3 = (-x + 2x) + (-8 + 3) = x - 5$$ 9. **Since the degree of the remainder $x - 5$ is less than the divisor's degree, we stop.** 10. **Write the final answer:** $$\frac{2x^{3}+3x^{2}+5x-8}{x^{2}+2x+3} = 2x - 1 + \frac{x - 5}{x^{2}+2x+3}$$